Finding matrix of transformation? Given the linear transformation: $T(M) = \begin{bmatrix}1&0\\0&2\end{bmatrix}M - M\begin{bmatrix}1&0\\0&2\end{bmatrix}$
a) Find the matrix $B$ of $T$ w/ respect to the standard basis $B$ of $\mathbb{R}^{2 \times 2}$.
b) Find bases of the image and kernel of $B$?
I'm stuck on part a) - I tried to plug in:
$M = \begin{bmatrix}1&0\\0&1\end{bmatrix}$
and find $T(M)$, but this just gives me the zero matrix. How would I do a)?
 A: Let 
$$
M=
\begin{bmatrix}
a&b\\c&d
\end{bmatrix}
$$
from the definition of $T $ we have:
$$
T(M)=\begin{bmatrix}
0&-b\\c&0
\end{bmatrix}
$$
So, representing $M$ and $T(M)$ as vectors in standard basis, $T$ acts as:
$$
\begin{bmatrix}
a\\b\\c\\d  
\end{bmatrix}
\quad \rightarrow \quad \begin{bmatrix}
0\\-b\\c\\0 
\end{bmatrix}
$$
a simple inspection shows that $T$ is represented by the matrix:
$$
\begin{bmatrix}
0&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&0 
\end{bmatrix}
$$
now you can find image and kernel.
A: In this case you will have to use a coordinatization of the space of matrices $\mathcal{M}_{2\times 2}(\mathbb{R})$ as $\mathbb{R}^4$. The usual coordinatization goes via $\begin{bmatrix}a & b\\ c & d\end{bmatrix}\mapsto [a,b,c,d]$.
In this fashion, you will have to plug in four matrices: $\begin{bmatrix}1 & 0\\ 0 & 0\end{bmatrix},\begin{bmatrix}0 & 1\\ 0 & 0\end{bmatrix},\begin{bmatrix}0 & 0\\ 1 & 0\end{bmatrix},\begin{bmatrix}0 & 0\\ 0 & 1\end{bmatrix}$. These matrices are the standard basis in $\mathcal{M}_{2\times 2}(\mathbb{R})$, and note that they correspond to the canonical base $\{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)\}$ according to our coordinatization.
Each of the operations will give you as an answer $2\times 2$ matrix, that you will have to transform in a vector using coordinatization again. The matrix of $T$ with respect to $B$ will then by a $4\times 4$ matrix.
From there, I think you can calculate Kernel and Image... just have always in mind that your final answers have to be given in terms of matrices, so you will have to use again the coordinatization (this time, to transform vector of $\mathbb{R}^4$ in matrices $2\times 2$).
